A319458 Decimal expansion of the location of the minimum of the error term E(x) in exp(x) * x^(-x) * Pi^(-1/2) * Gamma(1 + x) = (8*x^3 + 4*x^2 + x + E(x))^(1/6) for x > 0. A319459 provides the corresponding value of E.

6, 7, 1, 5, 0, 3, 7, 6, 5, 7, 6, 8, 0, 2, 5, 3, 6, 0, 8, 6, 4, 8, 1, 2, 0, 5, 7, 5, 4, 0, 2, 3, 0, 0, 3, 4, 7, 3, 5, 0, 3, 2, 0, 7, 0, 1, 8, 0, 6, 0, 8, 1, 8, 3, 6, 5, 8, 3, 0, 8, 0, 4, 4, 8, 0, 3, 6, 3, 0, 8, 8, 5, 5, 9, 0, 6, 0, 1, 6, 4, 9, 5, 3, 9, 6, 8, 4, 1, 3, 5

Offset

0,1

Comments

Ramanujan's question 754 in the Journal of the Indian Mathematical Society (VIII, 80) asked "Show that exp(x) * x^(-x) * Pi^(-1/2) * Gamma(1 + x) = (8*x^3 + 4*x^2 + x + E)^(1/6), where E lies between 1/100 and 1/30 for all positive values of x".

A numerical search provides an approximate minimum of E = 0.010045071877... (A319459) at x = 0.6715..., confirming Ramanujan's lower bound.

Links

Table of n, a(n) for n=0..89.

B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56, DOI: 10.1090/conm/236 (see Q754, JIMS VIII).

B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q754, JIMS VIII).

Example

0.6715037657680253608648120575402300347350320701806081836583...

Crossrefs

Cf. A319459.

Sequence in context: A340926 A356562 A011483 * A296459 A354684 A308915

Adjacent sequences: A319455 A319456 A319457 * A319459 A319460 A319461

Keyword

nonn,cons

Author

Hugo Pfoertner, Sep 19 2018

Status

approved